How to increase performance for GPU rendered interface parts?
Playback in the viewport is completely dependent on the CPU, unless you're in rendered mode. The CPU is used to calculate stuff like where your meshes are and how the armatures make them move. 

The graphics card is only used for rendering related tasks like calculating light bounces, shading, and colors. If you're in the solid mode of the viewport your graphics card only has to figure out what shade of grey the faces on your mesh are, which is a trivial task even for very weak GPUs.

Going back to CPUs, it's important to know that Blender's viewport is single-threaded. This means that if you have a quad-core CPU, the viewport can only use one of the four cores. If you look at the usage for individual cores rather than overall usage, you'll probably see one is maxed out and the rest are idle (in Windows task manager, right-click the CPU graph then select logical cores).

Here are a few ways that you might be able to increase your single-core performance besides upgrading your CPU:

1) Turn off Hyperthreading if your CPU has it. Hyperthreading is where your computer sees each physical CPU core as two (or more, but usually two) cores. This provides better performance for multi-threaded tasks like video editing, but effectively halves performance for single-threaded stuff like Blender's viewport. The option to turn it off is probably somewhere in your BIOS, which is unique to your computer so probably Google it. Some laptops (including mine >:( ) don't have the option to turn it off, so that stinks.

2) Clean out all the dust in your computer. When your computer gets too hot, it will automatically decrease performance so it doesn't overheat. This is called thermal throttling. If you clean out the dust with some compressed air, the heat sinks and fan will get better airflow and be able to keep your computer cool at higher speeds. This method is mostly aimed at laptops, since desktops tend to have much better cooling systems and don't really suffer from this.

3) Replace the thermal compound between your CPU and heatsink. Same idea as the last one, if your CPU can conduct heat more efficiently onto the heatsink, it'll throttle less and run faster. Stock thermal compounds tend to be pretty bad and can wear out. I'd suggest Arctic Silver 5.

4) Use Linux!! Linux handles the CPU much more efficiently than Windows, and you can get a pretty good performance increase just from switching over. For awhile I thought this was just Linux fanboys telling tall tales, but I recently switched my mega project over to Linux for RAM related reasons and I really felt the performance increase.

5) (edited in, somehow I forgot this on the original post)
Overclock if your PC supports it (but use caution).

Hope this helps :)